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Improved Quantitative Regularity for the Navier–Stokes Equations in a Scale of Critical Spaces

Archive for Rational Mechanics and Analysis volume 242, pages 1479–1531 (2021)Cite this article

Abstract

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier–Stokes equations satisfying certain critical conditions. The solutions we consider have \(\Vert r^{1-\frac{3}{q}}u\Vert _{L_t^\infty L_x^q}<\infty \) where \(r=\sqrt{x_1^2+x_2^2}\) and either \(q\in (3,\infty )\), or u is axisymmetric and \(q\in (2,3]\). Using the strategy of Tao (Quantitative bounds for critically bounded solutions to the Navier–Stokes equations. arXiv:1908.04958, 2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.

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Fig. 1

Notes

  1. We thank the anonymous referee for these references.

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Acknowledgements

The author is grateful to Terence Tao for many helpful discussions, as well as for providing feedback on a version of the manuscript. We also thank the anonymous referee for their careful reading and comments. This work was partially funded by NSF Grant DMS-1764034.

Funding

This work was partially funded by NSF Grant DMS-1764034.

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  1. UCLA Department of Mathematics, 520 Portola Plaza, Los Angeles, CA, 90025, USA

    Stan Palasek

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Correspondence to Stan Palasek.

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Communicated by P. Constantin

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Palasek, S. Improved Quantitative Regularity for the Navier–Stokes Equations in a Scale of Critical Spaces. Arch Rational Mech Anal 242, 1479–1531 (2021). https://doi.org/10.1007/s00205-021-01709-5

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